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1). \[\lim_{(x,y) \rightarrow (0,0)} \frac{ x ^{2}y ^{2} }{ x ^{4}+y ^{4} }\]
2). \[\lim_{(x,y) \rightarrow (0,0)} \frac{ x ^{2}y ^{2} }{ 2x ^{4}+y ^{4} }\]
In the first limit I used that \[f(x,y)=f(x,0) \rightarrow 0\] and \[f(x,y)=f(0,y)\rightarrow0\]
But in the second one that becomes totally wrong, since the xaxis goes to zero and yaxis to 1/3. Why should one change from f(o,y)> 0 to x=y in the second one, how do I recognize when I should set x=y ?
 one year ago
 one year ago
1). \[\lim_{(x,y) \rightarrow (0,0)} \frac{ x ^{2}y ^{2} }{ x ^{4}+y ^{4} }\] 2). \[\lim_{(x,y) \rightarrow (0,0)} \frac{ x ^{2}y ^{2} }{ 2x ^{4}+y ^{4} }\] In the first limit I used that \[f(x,y)=f(x,0) \rightarrow 0\] and \[f(x,y)=f(0,y)\rightarrow0\] But in the second one that becomes totally wrong, since the xaxis goes to zero and yaxis to 1/3. Why should one change from f(o,y)> 0 to x=y in the second one, how do I recognize when I should set x=y ?
 one year ago
 one year ago

This Question is Closed

shubhamsrgBest ResponseYou've already chosen the best response.0
Isn't answer to 1st one 1/2 ?
 one year ago

experimentXBest ResponseYou've already chosen the best response.0
it's a double limit, you have to be a bit careful with it ... since there are multiple ways to come at (0,0), I think there is a theorem .... but currently I am busty posting my own question.
 one year ago

shubhamsrgBest ResponseYou've already chosen the best response.0
http://www.wolframalpha.com/input/?i=limit+%28x%5E2+y%5E2+%2F%28x%5E4+%2By%5E4%29%29+as+x%3E0+and+y%3E0 What does this mean ?
 one year ago

experimentXBest ResponseYou've already chosen the best response.0
says that limit does not exist, try y=x or y=x^2, if you don't get the same limit then limit does not exist.
 one year ago

frxBest ResponseYou've already chosen the best response.0
According to the answerkey to Adams Calculus the answer for the first one is 0 and the second one DNE
 one year ago

ZarkonBest ResponseYou've already chosen the best response.0
\[\lim_{(x,y) \rightarrow (0,0)} \frac{ x ^{2}y ^{2} }{ x ^{4}+y ^{4} }\] let \(y=mx\) then \[\lim_{x \rightarrow 0} \frac{ x ^{2}(mx) ^{2} }{ x ^{4}+(mx) ^{4} }\] \[\lim_{x \rightarrow 0} \frac{ m^2x ^{4} }{ x ^{4}(1+m ^{4}) }\] \[\lim_{x \rightarrow 0} \frac{ m^2 }{ 1+m ^{4} }=\frac{ m^2 }{ 1+m ^{4} }\] pick any real number \(m\)
 one year ago
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